Bulletin of the American Mathematical Society , October 2002, page 563 —

“… the study of symmetries of patterns led to… finite geometries….”

– David W. Henderson, Cornell University

This statement may be misleading, if not (see Part II below) actually false. In truth, finite geometries appear to have first arisen from Fano's research on axiom systems. See The Axioms of Projective Geometry by Alfred North Whitehead, Cambridge University Press, 1906, page 13.

Part II: Grid

For the story of how symmetries of patterns later did lead to finite geometries, see the diamond theorem.

From the Bulletin of the American Mathematical Society , October 2002, p. 563:

“To produce decorations for their weaving, pottery, and other objects, early artists experimented with symmetries and repeating patterns. Later the study of symmetries of patterns led to tilings, group theory, crystallography, finite geometries, and in modern times to security codes and digital picture compactifications. Early artists also explored various methods of representing existing objects and living things. These explorations led to… [among other things] computer-generated movies (for example, Toy Story ).”

"During his career, he was consulted by, among others,
the crime writer Patricia Cornwell, and the artist Damien Hirst
(who used his expertise when working on his 2007 pieceFor the Love of God, a platinum cast of a skull, encrusted with diamonds)."

Quilts made by Amish women in Pennsylvania,
such as this traditional center diamond,
reveal the makers’ keen sense of color and design.

Household handicrafts and heirlooms made by American women seen as precursors to modern art is one underlying thesis of “Amish Abstractions: Quilts from the Collection of Faith and Stephen Brown,” a provocative exhibit on view at the de Young Museum through June.

Curated by Jill D’Alessandro of the Fine Arts Museums of San Francisco, the show features about 50 full-size and crib quilts made between 1880 and 1940 in Pennsylvania and the Midwest during what experts consider the apex of Amish quilt-making production.

Faith and Stephen Brown, Bay Area residents who began collecting quilts in the 1970s after seeing one in a shop window in Chicago and being bowled over by its bold design, say their continued passion for the quilts as art is in part because they’re so reminiscent of paintings by modern masters like Mark Rothko, Josef Albers, Sol LeWitt and Ellsworth Kelly — but the fabric masterpieces came first.

“A happy visual coincidence” is how the Browns and D’Alessandro define the connection, pointing to the brilliance in color theory, sophisticated palettes and complex geometry that characterize both the quilts and paintings.

A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.

The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn't matter which order the sub-expressions are evaluated in.

A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing: ω ω doesn't terminate by any evaluation path; and (\x. y) (ω ω) terminates only by some evaluation paths but not by others.

But the untyped lambda calculus enjoys some compensation for this weakness. It's Turing complete! It can represent any computation we know how to describe. (That's the cash value of being Turing complete, not the rigorous definition. There is a rigorous definition. However, we don't know how to rigorously define "any computation we know how to describe.") And in fact, it's been proven that you can't have both. If a computational system is Turing complete, it cannot be strongly normalizing.

There is no connection, apart from the common reference to an elementary geometric shape, between the use of "diamond" in the above Church-Rosser sense and the use of "diamond" in the mathematics of (Cullinane's) Diamond Theory.

"The links are direct between the tautology out of the Burning Bush, that 'I am' which accords to language the privilege of phrasing the identity of God, on the one hand, and the presumptions of concordance, of equivalence, of translatability, which, though imperfect, empower our dictionaries, our syntax, our rhetoric, on the other. That 'I am' has, as it were, at an overwhelming distance, informed all predication. It has spanned the arc between noun and verb, a leap primary to creation and the exercise of creative consciousness in metaphor. Where that fire in the branches has gone out or has been exposed as an optical illusion, the textuality of the world, the agency of the Logos in logic—be it Mosaic, Heraclitean, or Johannine—becomes 'a dead letter.'"

"It is often interesting whether a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem (and the corresponding Church-Rosser confluence property)."

"Crucial here is the concept of emergence , the sudden coming into being of something that is greater than its component parts (like water is greater than its component parts, hydrogen and oxygen; therefore, water is an emergent property)."

“Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately). And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought. Both anecdotal and geometric, it builds coherent structures out of ‘the odds and ends left over from psychological or historical process.’

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore. (How, precisely, they have come into being in the first place is one of the points on which Levi-Strauss is not too explicit, referring to them vaguely as the ‘residue of events… fossil remains of the history of an individual or a society.’) Such images are inevitably embodied in larger structures– in myths, ceremonies, folk taxonomies, and so on– for, as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular. But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort. Quoting Franz Boas that ‘it would seem that mythological worlds have been built up, only to be shattered again, and that new worlds were built from the fragments,’ Levi-Strauss generalizes this permutational view of thinking to savage thought in general.”

"Thanks in part to this bunker mentality, American Christianity has become what Hunter calls a 'weak culture'— one that mobilizes but doesn’t convert, alienates rather than seduces, and looks backward toward a lost past instead of forward to a vibrant future. In spite of their numerical strength and reserves of social capital, he argues, the Christian churches are mainly influential only in the 'peripheral areas' of our common life. In the commanding heights of culture, Christianity punches way below its weight."

"Philosophy seeks not absolute first principles, nor yet purely immediate insights,
but the self-mediation of the system of truth, and an insight into this self-mediation.
Axioms, in the language of modern theory, are best defined, neither as certainties
nor as absolutely first principles, but as those principles which are used as the first
in a special theory.

LITERATURE — A complete view of the literature of the problems
regarding axioms is impossible, since the topic is connected with all
the fundamental philosophical issues…. JOSIAH ROYCE"

“I’ve got two Broadway shows, a feature film, and Mozart,’’ she said.
“It’s a very interesting place to be and to be able to move back and forth,
but at a certain point you have to be able to step outside and see,’’
and here she dropped her voice to a tranquil whisper, “it’s just theater.
It’s all theater. It’s all theater. The whole thing is theater.’’

Wednesday, December 15, 2010

"It’s very tricky choosing a rich and interesting case study which is philosophically salient. To encourage the reader or listener to follow up the mathematics to understand what you’re saying, there must be a decent pay-off. An intricate twentieth century case study had better pack plenty of meta-mathematical punch."

Tuesday, December 14, 2010

My principal meditations for some time have been directed towards
the application of the theory of ambiguity to transcendental
analysis. It was a question of seeing a priori in a relation
between quantities or transcendent functions, what exchanges one
could make, which quantities one could substitute for the given
quantities without the original relation ceasing to hold. That
immediately made clear the impossibility of finding many expressions
that one could look for. But I do not have time and my ideas are
not yet well developed on this ground which is immense.

“I’ve got two Broadway shows, a feature film, and Mozart,’’ she said. “It’s a very interesting place to be and to be able to move back and forth, but at a certain point you have to be able to step outside and see,’’ and here she dropped her voice to a tranquil whisper, “it’s just theater. It’s all theater. It’s all theater. The whole thing is theater.’’

"Symmetry plays an important role in geometry, number theory, and quantum physics. I will discuss the links between these areas from the vantage point of the Langlands Program. In this context 'duality' means that the same theory, or category, may be described in two radically different ways. This leads to many surprising consequences."

“I carry the past inside me like an accordion, like a book of picture postcards that people bring home as souvenirs from foreign cities, small and neat,” she wrote in her memoir. “But all it takes is to lift one corner of the top card for an endless snake to escape, zigzag joined to zigzag, the sign of the viper, and instantly all the pictures line up before my eyes.”

Wednesday, December 8, 2010

After the final no there comes a yes
And on that yes the future world depends.
No was the night. Yes is this present sun.
If the rejected things, the things denied,
Slid over the western cataract, yet one,
One only, one thing that was firm, even
No greater than a cricket's horn, no more
Than a thought to be rehearsed all day, a speech
Of the self that must sustain itself on speech,
One thing remaining, infallible, would be
Enough. Ah! douce campagna of that thing!
Ah! douce campagna, honey in the heart,
Green in the body, out of a petty phrase,
Out of a thing believed, a thing affirmed:
The form on the pillow humming while one sleeps,
The aureole above the humming house . . .

For a fanciful linkage of the dreidel 's concept of chance
to The Stone 's concept of invariant law, note that the
New York Lottery evening number on Dec. 1 (the
beginning of Hanukkah) was 840. See also the number
840 in the final post (July 20, 2002) of a search forSolomon's Cube.

“The human mind is comforted by the notion that a greater power has all the answers. A world in which secret messages swirl around us is exciting because it helps reinforce our belief that true enlightenment is within our reach.”

With a nod to film-maker Stan Brakhage, Davenport calls his compositional principle "architectonic form." (2) In the essay "Narrative Tone and Form," he identifies in twentieth century literature "a movement from assuming the world to be transparent, and available to lucid thoughts and language, to assuming (having to assume, the artists involved would say) that the world is opaque" (Geography 311). Architectonic form derives from modernist experiments in disrupted perspective (as, for example, in collage and vorticism). "The architectonics of a narrative," Davenport says, "are emphasized and given a role to play in dramatic effect when novelists become Cubists; that is, when they see the possibilities of making a hieroglyph, a coherent symbol, an ideogram of the total work. A symbol comes into being when an artist sees that it is the only way to get all the meaning in. Genius always proceeds by faith" (312). The unparaphrasable architectonic text "differs from other narrative in that the meaning shapes into a web, or globe, rather than along a line" (318). The essence of such art "is that it conceals what it most wishes to show; first, because it charges word, image and sense to the fullest, fusing matter and manner; secondly, to allow meaning to be searched out" (57-58).

In architectonic form, meaning may be generated more in the interstices between images, citation, and passages of dialogue than in the content of these elements. "It is the conjunction, not the elements, that creates a new light," Davenport says in an essay on poet Ronald Johnson (194). This is the Poundian aesthetic Charles Olson attempted to translate into practical pedagogical terms as rector of Black Mountain College, a school organized, as Olson explained in a 1952 letter, on the "principle that the real existence of knowledge lies between things & is not confined to labeled areas" (quoted in Duberman 341).

Note:

(2) Brakhage has written admiringly of Davenport. In "Ice is for Coffee and for Wine" he speaks of the joy to have "at last met such a man as Pound describes Remy de Goncourt to have been…i.e. one whose intelligence was a way of feeling" (7).

Fucking Hell is not, evidently, a realistic (much less nit-picking ) account of the ….
The following link enables you to pan virtually around the Bourbaki…
www.whitecube.com/artists/chapman/texts/154/ – Cached

CBR: “Big Time” is this title of this new era of “Amazing Spider-Man.” Why choose that title? What exactly is it referring to?

DAN SLOTT: “Big Time” refers to more than “Amazing Spider-Man,” it also refers to other Spider-Projects: “Astonishing Spider-Man/Iron Man,” the new Norman Osborn mini, and the all-new “Spider-Girl!” With “Amazing,” “Big Time” takes on a lot of meanings. In this book, everything is bigger: bigger stakes for Peter Parker, bigger threats for Spider-Man, and a much bigger comic. We are expanding to 30 pages of material, twice a month!

The holiday is celebrated with, among other things, the Jewish version of a die— the dreidel. Note the similarity of the dreidel to an illustration of The Stone* on the cover of the 2001 Eerdmans edition of Charles Williams's 1931 novel Many Dimensions—

For mathematics related to the dreidel , see Ivars Peterson's column on this date fourteen years ago.
For mathematics related (if only poetically) to The Stone , see "Solomon's Cube" in this journal.

Here is the opening of Many Dimensions—

For a fanciful linkage of the dreidel 's concept of chance to The Stone 's concept of invariant law, note that the New York Lottery yesterday evening (the beginning of Hanukkah) was 840. See also the number 840 in the final post (July 20, 2002) of the "Solomon's Cube" search.

Some further holiday meditations on a beginning—

Today, on the first full day of Hanukkah, we may or may not choose to mark another beginning— that of George Frederick James Temple, who was born in London on this date in 1901. Temple, a mathematician, was President of the London Mathematical Society in 1951-1953. From his MacTutor biography—

"In 1981 (at the age of 80) he published a book on the history of mathematics. This book 100 years of mathematics (1981) took him ten years to write and deals with, in his own words:-

those branches of mathematics in which I had been personally involved.

He declared that it was his last mathematics book, and entered the Benedictine Order as a monk. He was ordained in 1983 and entered Quarr Abbey on the Isle of Wight. However he could not stop doing mathematics and when he died he left a manuscript on the foundations of mathematics. He claims:-

The purpose of this investigation is to carry out the primary part of Hilbert's programme, i.e. to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced."

"… one might crudely distinguish between philosophical and mathematical motivation. In the first case one tries to convince with a telling conceptual story; in the second one relies more on the elegance of some emergent mathematical structure. If there is a tradition in logic it favours the former, but I have a sneaking affection for the latter. Of course the distinction is not so clear cut. Elegant mathematics will of itself tell a tale, and one with the merit of simplicity. This may carry philosophical weight. But that cannot be guaranteed: in the end one cannot escape the need to form a judgement of significance."

Here Hyland appears to be discussing semantic ("philosophical," or conceptual) and syntactic ("mathematical," or structural) approaches to proof theory. Some other remarks along these lines, from the late Gian-Carlo Rota—

* Williams's novel says the letters of The Stone are those of the Tetragrammaton— i.e., Yod, He, Vau, He (cf. p. 26 of the 2001 Eerdmans edition). But the letters on the 2001 edition's cover Stone include the three-pronged letter Shin , also found on the dreidel . What esoteric religious meaning is implied by this, I do not know.

For some philosophical background on illusion and reality, see Graham Priest in the Sunday, Nov. 28, New York Times column "The Stone" and in a work of fiction he published in the Notre Dame Journal of Formal Logic (Vol. 38, No. 4) in 1997.

Wagner star Peter Hofmann died yesterday, November 30. The above obituary,
together with the NY Lottery numbers for the end of November and
the remarks on Nietzsche in this journal Sunday morning, suggest
the following readings from Nietzsche's The Will to Power—

Introduction (for National Novel Writing Month)—

"He [Nietzsche's "great man"] would rather lie than tell the truth,
because lying requires more spirit and will."

Details (according to the lottery numbers)—

For deeper background, see the life of the Will to Power translator, Anthony Ludovici.